Solving For X: F(x) = 2x + 3, F(x) = 11
Hey guys! Today, we're diving into a super basic but essential algebra problem. We've got the function f(x) = 2x + 3, and we need to figure out what x is when f(x) equals 11. Don't worry; it's easier than it sounds! Let's break it down step by step. Understanding functions is super important in math, and this is a great starting point.
Understanding the Function
First, let's make sure we all understand what a function is. Think of a function like a machine. You put something in (that's our x), and the machine does something to it and spits out a result (that’s our f(x)). In this case, our machine (f(x)) takes x, multiplies it by 2, and then adds 3. So, if we put in x = 1, the machine would do 2 * 1 + 3, which equals 5. That means f(1) = 5. Got it? This concept is fundamental not just in algebra but also in calculus and many other areas of math. Being comfortable with functions will really help you out later on!
Setting Up the Equation
Now, here's the cool part. We know that f(x) is supposed to be 11. That means whatever our machine does to x, we need to end up with 11. So we can write this as an equation: 2x + 3 = 11. This equation is the key to solving for x. It tells us that if we double x and then add 3, we should get 11. This is a simple algebraic equation that we can easily solve by isolating x. Remember that our goal is to get x all by itself on one side of the equation so we can see what its value is. So, let's get to it!
Solving for x
Alright, let's solve for x. Remember, our equation is 2x + 3 = 11. The first thing we want to do is get rid of that + 3 on the left side. We can do that by subtracting 3 from both sides of the equation. Why both sides? Because we need to keep the equation balanced! If we do something to one side, we have to do the exact same thing to the other side to make sure they stay equal. So, if we subtract 3 from both sides, we get:
2x + 3 - 3 = 11 - 3
This simplifies to:
2x = 8
Now we're getting closer! We have 2x = 8. That means 2 times x equals 8. To find out what x is, we need to get rid of that 2 that's multiplying it. We can do that by dividing both sides of the equation by 2. Again, we do it to both sides to keep things balanced:
(2x) / 2 = 8 / 2
This simplifies to:
x = 4
The Answer
Boom! We did it! We found that x = 4. That means when x is 4, f(x) equals 11. We can double-check this by plugging x = 4 back into our original function: f(4) = 2 * 4 + 3 = 8 + 3 = 11. Yep, it works! So, the answer to the question "If f(x) = 2x + 3 and f(x) = 11, what is x?" is x = 4. Understanding how to solve these types of problems is crucial for success in algebra and beyond, so pat yourselves on the back for working through it!
Why This Matters
You might be thinking, "Okay, cool, we solved for x. But why does this even matter?" Well, solving for variables in equations is a fundamental skill in mathematics and has tons of applications in real life. For example, imagine you're running a small business, and you want to figure out how many products you need to sell to make a certain amount of profit. You could set up an equation just like this one, where x represents the number of products, and f(x) represents your profit. By solving for x, you can figure out exactly how many products you need to sell to reach your profit goal.
Or, let's say you're planning a road trip and want to know how long it will take to get to your destination. You could use a function to relate the distance you need to travel to the time it will take, taking into account your average speed. By solving for the time variable, you can estimate how long your trip will take. The possibilities are endless! The ability to solve equations is a powerful tool that can help you make informed decisions in all sorts of situations.
Practice Makes Perfect
Okay, now that we've solved this problem together, it's time for you to practice on your own. Here are a few similar problems you can try:
- If f(x) = 3x - 2 and f(x) = 10, what is x?
- If f(x) = x / 2 + 5 and f(x) = 8, what is x?
- If f(x) = -2x + 7 and f(x) = 1, what is x?
Work through these problems step by step, just like we did with the original one. Remember to isolate x by performing the opposite operations on both sides of the equation. Don't be afraid to make mistakes – that's how we learn! The more you practice, the more comfortable you'll become with solving these types of problems. And remember, if you get stuck, you can always ask for help from a teacher, tutor, or friend. Keep practicing, and you'll be solving equations like a pro in no time!
Wrapping Up
So, there you have it! We successfully solved for x when given the function f(x) = 2x + 3 and the value f(x) = 11. We learned about functions, setting up equations, and isolating variables. Remember, the key to success in algebra is understanding the basic concepts and practicing regularly. Keep up the great work, and you'll be well on your way to mastering algebra! You got this!
And hey, if you found this helpful, let me know! Maybe I'll do some more math problems. Until next time, happy solving!